JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX

YUSUF HARTONO

COR KRAAIKAMP

FRITZ SCHWEIGERAlgebraic and ergodic properties of a new continued fractionalgorithm with non-decreasing partial quotientsJournal de Théorie des Nombres de Bordeaux, tome 14, no 2 (2002),p. 497-516<http://www.numdam.org/item?id=JTNB_2002__14_2_497_0>

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497-

Algebraic and ergodic properties of a newcontinued fraction algorithm withnon-decreasing partial quotients

par YUSUF HARTONO, COR KRAAIKAMPet FRITZ SCHWEIGER

RÉSUMÉ. On introduit la notion de développement en fractionscontinues de Engel. Nous étudions notamment les propriétés er-godiques de ce développement et le lien avec celui introduit parF. Ryde monotonen, nicht-abnehmenden Kettenbruch.

ABSTRACT. In this paper the Engel continued fraction (ECF) ex-pansion of any x ~ (0,1) is introduced. Basic and ergodic proper-ties of this expansion are studied. Also the relation between theECF and F. Ryde’s monotonen, nicht-abnehmenden Kettenbruch(MNK) is studied.

1. Introduction

Over the last 15 years the ergodic properties of several continued fractionexpansions have been studied for which the underlying dynamical systemis ergodic, but for which no finite invariant measure equivalent to Lebesguemeasure exists. Examples of such continued fraction expansions are the’backward’ continued fraction (see ~AF~ ), the ’continued fraction with evenpartial quotients’ (see [S3]) and the Farey-shift (see [Leh]). All these (andother) continued fraction expansions are ergodic, and have a a-finite, infi-nite invariant measure. Since these continued fractions are closely relatedto the regular continued fraction (RCF) expansion, their ergodic proper-ties follow from those of the RCF by using standard techniques in ergodictheory (see [DK]).

In this paper we introduce a new continued fraction expansion, which wecall-for reasons which will become apparent shortly-the Engel continuedfraction (ECF) expansion. We will show that this ECF has an underlyingdynamical system which is ergodic, but that no finite invariant measureequivalent to Lebesgue measure exists for the ECF.As the name suggests, the ECF is a generalization of the classical Engel

Manuscrit re~u le 27 septembre 2001.

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Series expansion, which is generated by the map S’ : [0, 1) -+ [0, 1), givenby

I ’B.

where [g J is the largest integer not exceeding ~, see also Figure 1.

FIGURE 1. Engel Series map S

Using S, one can find a (unique) series expansion of every x E (o,1 ) ,given by

1 1 1

where qn = + 1, n &#x3E; 1. In fact it was W. Sierpinski[Si] in 1911 who first studied these series expansions.The metric properties of the Engel Series expansion have been studied

in a series of papers by E. Borel [B], P. Levy [L], P. Erd6s, A. R6nyi andP. Sz3sz [ERS] and R6nyi [R]. In [ERS] it is shown that the random vari-ables Xl = logql, Xn = log(qn/qn-i) are ’almost independent’ and ‘almostidentically distributed’. From this first the central limit theorem is derivedfor log qn, then the strong law of large numbers and finally the law of theiterated logarithm are obtained. The second result has been announcedearlier without proof by E. Borel [B]. The first and third results are dueto P. Levy [L]. In [R], R6nyi finds new (and more elegant) proofs to theseand other results. Later F. Schweiger [Sl] showed that S is ergodic, andM. Thaler [T] found a whole family of Q-finite, infinite measures for S.Further information on the Engel Series (and the related Sylvester Series)can be found in the books by J. Galambos [G] and Schweiger [S3].

Clearly one can generalize the Engel Series expansion by changing thetime-zero partition, or by ’flipping’ the map S on each partition element(thus obtaining an alternating Engel Series expansion, see also [K2K]).

499

Let be a monotonically decreasing sequence of numbers in (o,1),with rl = 1, rn &#x3E; 0 for n &#x3E; 1 and 0. Furthermore, let

en = 1, and let the time-zero partition P be given by

Then

where n E N is such that x E [fn) rn) generalizes the ’Engel-map’ S. Withsome effort the ideas from the above mentioned papers can be carried overto this generalization, also see W. Vervaat’s thesis (V).

In this paper we study a different variation of the Engel Series expansion.Let the Engel continued fraction (ECF) map TE : [0, 1) - [0, 1) be givenby

see also Figure 2. Notice that

where T : [0, 1] ~ [0, 1) is the regular continued fraction map, given by

and = For any x E (0,1), the ECF-map ’generates’ (in a way

FIGURE 2. The ECF-map TE and the RCF-map T

500

similar to the way the RCF-map T ’generates’ the RCF-expansion of x) anew continued fraction expansion of x of the form

In this paper we will study in Section 3 the ergodic properties of this newcontinued fraction expansion, the so-called Engel continued fraction (ECF)expansion. However, since the ECF is new, we will first show in the nextsection that the ECF ’behaves’ in many ways like any other (semi-regular)continued fraction. In the last section we will study the relation betweenthe ECF, and a continued fraction expansion introduced by F. Ryde [Ryl]in 1951, the so-called monotonen, nicht-abnehmenden Kettenbruch (MNK).We will show that the ECF and a minor modification of Ryde’s MNK aremetrically isomorphic, and due to this many properties of the ECF-suchas ergodicity, the existence of a-finite, infinite measures-can be carriedover to the MNK. Conversely, the fact that not every quadratic irrational xhas an ultimately periodic MNK-expansion can be carried over to the ECFvia our isomorphism.

2. Basic properties

In this section we will study the basic properties of the ECF. In manyways it resembles the RCF, but there are also some open questions whichsuggest that there are fundamental differences.

Let x E (0, ), and define

From definition (1) of TE it follows that

where T# (z ) = z and = for n &#x3E; 1, and one has-similarto the Engel Series case-that

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As usual the convergents are obtained via finite truncation;

The finite continued fraction in the left-hand side of (5) is denoted by[[0; b1,... , bn]].

It is clear that the left-hand side of (5) is a rational number, so that wehave the following result.

Theorem 2.1. Let x E (0, 1), then x has a finite ECF-expansion (i.e.,= 0 for some n &#x3E; 1) if and if x E Q.

Proof. The necessary condition is obvious since x in (5) is a rationalnumber. For the sufficient condition, let x = P/Q with P, Q E N and0 P Q. Then

where bl = LQ/PJ. It is clear that 0 f Q - b1P P and 1 ~ Qbecause Q = b1P + r with 0 ~ r P by the Euclidean algorithm. Now let

then

Since p(N) E N U 101 there exists an n, &#x3E; 1 such that TE(x) = 0. Noticethat one has to apply TE to x at most P times to get TË(x) = 0. 0

The proof of the following theorem is omitted, since it is quite straight-forward. For a proof the interested reader is referred to Section 1 in [K],where a similar result has been obtained for a general class of continuedfractions.

Proposition 2.1. Let the sequences and (Qn)n&#x3E;o recursively bedefined by

502

Then and are both increasing sequences. Furthermore,one has for n &#x3E; 1 that

-

-

and

For the RCF-expansion it is known that even-numbered convergents arestrictly increasing and odd-numbered strictly decreasing and that everyeven-numbered convergent is less than every odd-numbered one. The sameresult also holds for ECF convergents as stated in the following proposition.

Proposition 2.2. Let Cn = Pn/Qn be the n-th ECF-convergent of x E[0,1)BQ. Then

Moreover, C2j C2k+l for any nonnegative j and k.

Proof. It follows from Proposition 2.1, using (6), that

Dividing both sides by QnQn-2 gives

Since b j &#x3E; 1 for all j (and so are the Qj’s), C~ - &#x3E; 0 if n iseven. Hence, C2m-2 C2,.,1 for 1. Similarly, C2m+l Now

upon division on both sides of (7) by one obtains Cn-1 =which gives Cn_ 1 Cn if n is odd, that is,

C2m Thus for any nonnegative j and k

as desired. 0

In the next proposition, we will see that the sequence of ECF-convergentsconverges to the number from which it is generated.

Proposition 2.3. For x E ~0,1), let be the sequence of ECF-convergents of x. Then Pn/Qn = x.

-

Proof. In case x is rational, the result is clear; see Theorem 2.1. Now

suppose that x is irrational. By induction one has that

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In fact, if x is rational the special case Tp(x) = 0 gives x = Pn~Qn. From(7) and (8) it follows that

which trivially yields that

Letting ~n = and using (6), one can see that ~n is one of the termsin QnQn+1 so that the right-most side of (10) goes to zero as n - oo. Thiscompletes the proof. D

Notice that (9) yields that CZn ~ C2n+1, for n &#x3E; 1, where Cn =

Pn/Qn. We now will show that the ECF-expansion is unique.

Proposition 2.4. Let be a sequence of positive integers satisfying(4), and let the sequences and given by (6). Then thelimit

exists. Say this limit equals x, then x E (0, 1). Furthermore, bn = bn(x) forn &#x3E; l.

Proof. Let Cn = Pn/Qn. From the proof of Proposition 2.2 it followsthat (C2~ ) is an increasing sequence that is bounded from above; there-fore, C2k exists. Similarly, is a decreasing sequence thatis bounded from below, and so also exists. It remains toshow that these two limits are equal. To this end, note that, in the proofof Proposition 2.2, C2k - 0 as k --~ oo (by the same argument asin the proof of Proposition 2.3) so that liMk,,,,, Letx = C2k; then Ck exists and equals x. This completes thefirst part of the proof.

Since 0 C2 x 1, the second statement that x E (o,1) followstrivially.

For the third part, recall that by definition of Cn we have for n &#x3E; 2

Setting for n &#x3E; 2

it follows from the first part of the proof that there exists a number x E(0, 1) such that x. By letting n - oo in (11) we find that

504

from which

Since bi (z) is the unique positive integer for which x - E ~0,1), itfollows that b, (x) is also the unique positive integer for which

But then it follows from (12) that b1(x) = bl, and moreover we see thatTE(x) = Repeating the above argument, now applied to TE(x), yieldsthat b2 (x) = b2. By induction one finds that = bn for n E N. 0

Let x E [0, 1) B Q. We denote the (infinite) ECF-expansion (2) of x by

For the RCF-expansion one has the theorem of Lagrange, which statesthat x is a quadratic irrational (i.e., x E and is a root of ax2 + bx + c =0, a, b, c E Z) if and only if x has a RCF-expansion which is ultimately(that is, from some moment on) periodic. For the ECF the situation ismore complicated. Suppose x E ~0,1) has a periodic ECF-expansion, say

where the bar indicates the period. Clearly one has that is a quadraticirrational, and from (4) it follows that the period-length t is always equalto 1.

Due to this, purely periodic expansions can easily be characterized; forn E N one has

One could wonder whether every quadratic irrational x has an eventuallyperiodic ECF-expansion. In [Ry2], Ryde showed that a quadratic irrationalx has an eventually periodic MNK-expansion if and only if a certain set ofconditions are satisfied. In Section 4 we will see that the ECF-map TE anda modified version of the MNK-map are isomorphic, and due to this wewill obtain that not every quadratic irrational x has an eventually periodicECF-expansion.An important question is the relation between the convergents of the

RCF and those of the ECF. Let x E [0, 1) B Q, with RCF-expansion x =[0;ai,a2?..’]) with RCF-convergents and ECF-convergents

Moreover, define the mediant convergents of x by

505

then the question arises whether infinitely many RCF-convergents and/ormediants are among the ECF-convergents, and conversely.

mediants.

A related question is the value of the first point in a ’Hurwitz spectrum’for the ECF. Let x E [0, 1) B Q, again with RCF-convergents and

ECF-convergents where we moreover assume that

1 for n &#x3E; 0. Setting for n &#x3E; 0

one has the classical results that

which trivially implies that

infinitely often for all x E (0,1) B Q.

If infinitely many RCF-convergents of x are also ECF-convergents of x,then one has that

(14) C infinitely often,

with C = 1. Consider the number x having a purely periodic expansionwith bn = 2. Then x = !( -1 + v’3). The difference equation An = +

2An-2 controls the growth of QnlQnx - Pnl. Its eigenvalues are1 - y’3 and 1 + y’3, and therefore we see that 8n(x) = [ is

asymptotically equal to 21. Furthermore from (9) one can see that bn+1+2 1 b

Pni which shows that such a constant C

cannot exist for all x.

3. Ergodic propertiesIn this section we will show that TE has no finite invariant measure,

equivalent to the Lebesgue measure A, but that TE has infinitely manya-finite, infinite invariant measures. Furthermore it is shown that TE is

ergodic with respect to A.

506

Let

and define for n E N, bl, ... , bn E N with ... bn the cylinder sets(or: fundamental intervals) B (bi, ... , bn ) by

Then it is clear that, see also Figure 2,

Now let p be a finite TE-invariant measure, that is,

for any Borel set A E (0,1).Since it is a measure, we have that

where Tn E B(n) denotes the invariant point under TE, see also (13). Onthe other hand,

because IL is TE-invariant. Hence, since we assumed that ti is a finite

measure,

Furthermore, we also have

Similar arguments yield that ~([3/5, ri]) = p([8 /13, Ti]), and therefore

so that we have

Continue this iteration to see that B(1) must have its mass con-

centrated at Ti. Next, it follows from (16) that TE1B(2) _ (1/3,3/8] U(2/3,3/4]. ° But p,(2/3, 3/4] = 0 so that ~(T~B(2)) = ¡.¿((1/3, 3/8]) and fol-lowing the same arguments as above gives that B(2) has its mass /~(.B(2))concentrated at 72.

507

Applied to other values of n, induction yields that on (0, 1) the measurep has mass p(B(n)) concentrated at Tn for n &#x3E; 1. Consequently, we haveproved the following result.

Theorem 3.1. There does not exist a non-atomic finite TE-invariant mea-sure.

Next, we will prove ergodicity of TE with respect to Lebesgue measure.

Theorem 3.2. TE is ergodic with respect to Lebesgue measure A.

Proof. Let TE lA = A be an invariant Borel set. Define

where cA denotes the indicator function of A. Then we calculate

We put

Note that

that

and that it follows from (8) and (7)

I .

From this it follows that

Moreover we find that

508

and that

which shows that

For n = 1 we have a more precise estimate. Since

we get

Together with (18) this yields that

Furthermore we have that

Note that for Engel’s series 6(b) = d(b) which fact makes the proof easier.

Together with (B) we get the estimate d(b + 1)

The Martingale Convergence Theorem shows that

almost everywhere,

see also Theorem 9.3.3 in [S3]. If = 1 a.e. there is nothing to show.Let us therefore assume that a(A) 1. Suppose that there is some z E Acsuch that = 0. Then for n sufficiently large8(bl (z), ... , b~,(z)) for any given e &#x3E; 0, and by (A) for b = bn(z) wefind d(b) Applying (D) this yields that d(c) 4 for all c &#x3E; b.

Since the set Fnr = N, j &#x3E; 1} is countable (since everyx E FN ends in a periodic ECF-expansion with digit b N), it follows thatFN has measure 0 and we clearly have bn = oo a.e. Therefore by(A) we see that 6(bi (z) , ... , bn(x)) ~ for almost all points x. Assumingthat e 1 this shows that cA(x) = 0 almost everywhere, i.e., A(A) = 0. 0

For the Engel’s series R6nyi [R] showed that for almost all x the sequenceof digits is monotonically increasing from some moment no(x) on. For theECF a similar result holds.

Theorem 3.3. For almost all x E (0,1) the sequence of digits is strictly increasing for some n &#x3E; no(x).

509

Proof. Setting y := Q. it follows from (17) and (6) that

and that 0 y 1If we put bn = bn+l we immediately get

Lemma 3.1.

Proof (of the lemma, see also [S3], p. 68-69). It follows from (19) that

equals

Therefore we have to estimate the sum

For b = a the first term while for b &#x3E; a + 1 we use

0 y C a to obtain the estimate

and it follows that

Using

510

we eventually get

This expression has its maximum for a = 1 which gives the value 313The claim on the maximal value can be seen as follows.

The sum of the four terms

decreases to 0 as a - oo and becomes smaller than .1 for a &#x3E; 3. On theother hand the remaining term

is increasing and is bounded by 2. Therefore numerical calculations sufficefor n = 1, 2, 3. This proves the Lemma.

Now the Borel-Cantelli lemma yields that the set of all points x for which= bn+1(x) for infinitely many values of n has measure 0. 0

Now we give two constructions of a-finite, infinite invariant measures forTE. The first construction follows to some extent Thaler’s constructionfrom [T] of Q-finite, infinite invariant measures for the Engel’s series mapS.

First, let B(n) be as in (15). Define

Obviously one has that A(n, k) fl A(m, e) = 0 for (n, k) # (m, e) and that,apart from a set of Lebesgue measure zero

We now choose a monotonically increasing sequence of positive real num-bers satisfying

For any non-purely periodic x E B = B(n), there exist positive in-tegers nand i such that x E A(). For any non-eventually periodicx E B B Q we inductively define a sequence by

511

and

For x E [0, 1) B Q given one has, since

and for k &#x3E; 2 one has

ax+ux) - = n(TE(x) + 1)2 (ak (TE (z) ) - O:k-l (TE (z) ) ) .So by induction it follows that the sequence is a positive mono-tonically non-decreasing sequence for each x E [0,1). We will show that

and h(x) := 0 for x fI. is a density of a measure equivalent toLebesgue measure, that is, it satisfies Kuzmin’s equation, see also Chapter13 in [83]. First note that for x E B(n) Kuzmin’s equation reduces to

where Vj is the local inverse of TE on B(j), i.e.,

As in Thaler’s case for the Engel Series expansion, this construction yieldsinfinitely many different cr-fmite, infinite invariant measures which are notmultiples of one-another.

For the second construction, let Note that

Furthermore, let G &#x3E; 1 be the golden mean, defined by G2 =

512

Here Note that go is a

solution of the functional equationSetting for t = 2, 3, ...

then h(x) := gt- 1 (x) on B(t) is an invariant density. To see this note thatKuzmin’s equation reads

on B(k). Then for x E B(k) we have on the left hand, whileexpanding the right hand side gives

In the calculation we used that =

We end this section with a theorem on the renormalization of the ECF-

map TE. See also Hubert and Lacroix [HL] for a recent survey of the ideasbehind the renormalization of algorithms.

Setting. , - I ..-

then clearly 0 1. We have the following theorem.

Theorem 3.4. Let 7 = 324, then

Proof. We introduce for t E [0, 1] the map

Applying the chain-rule we find, see also (8)

513

Then where

On the other hand we have, see also (17)

Therefore

which yields that

and

Remark. Following the ideas in Schweiger [S2] it should be easy to provethat the sequence is uniformly distributed for almost all points x.

4. On Ryde’s continued fraction with non-decreasing digitsIn 1951, Ryde [Ryl] showed that every x E (0, 1) can be written as a

monotones, nicht-abnehmenden Kettenbruch (MNK) of the form

which is finite if and only if x is rational.Underlying this expansion is the map SR : (0, 1) ~ (0,1), defined by

514

However, and this was already observed by Ryde, one has that

and therefore we might as well restrict our attention to the interval (2,1),and just consider the map TR : (1, 1) - (~ 1), given by

see also [S3], p. 26. Now every x E (- 1) has a unique NMK of the form(20) (with s = 1), which we abbreviate by

The following theorem establishes the relation between the ECF and theMNK. We omitted the proof, since it follows by direct verification.

Theorem 4.1. Let the bijection 0 : (0, 1) ~ (~? 1) be defined by

Then

Furthermore, for

and if we defines 1 the cylinders of TR by

then

Due to Theorem 4.1 we can ’carry-over’ the whole ’metrical structure’of the ECF to the MNK. To be more precise, letting x3 be the collection ofBorel sets of ( 2 1), and setting

where p is a a-finite, infinite TE-invariant measure on (0,1) with density h(with h from Section 3), then we have the following corollary.

Corollary 4.1. The map TR is ergodic with respect to Lebesgue measureA, but no finite TR-invariant measure exist equivalent to À. Each of themeasures v from (23) is a a-finite, infinite TR-invariant measure on (4,1).

515

Proof. We only give a proof of the first statement. Suppose that thereexists a Borel set A C (!,1) for which 0 A(A) A(, 1) = 1, suchthat Ti1(A) = A. From the fact that § : (0,1) -&#x3E; (~1) is a bijection,and due to (21) one has that ~-1 (A) is a TE-invariant set, and hencea(~-1(A)) E {0,1}, which is impossible. 0

To conclude this paper, let us return to the question of periodicity ofthe ECF-expansion of a quadratic irrational x. Due to (22) we have thatthe ECF-expansion of x E (0,1) is (ultimately) periodic if and only ifthe NMK-expansion of O(x) E (!,1) is (ultimately) periodic. Themain result of Ryde’s second 1951 paper [Ry2] now states that a quadraticirrational ~ E (0, 1) has an (ultimately) periodic NMK-expansion if andonly if a (rather large) set of constraints-too large to be mentioned here;the statement of his theorem covers almost 2 pages!-has been satisfied.Due to Theorem 4.1 these constraints can trivially be translated into a setof constraints for the ECF.As a consequence there exist (infinitely many) quadratic irrationals x for

which the ECF-expansion is not ultimately periodic. We end this paperwith an example.

Example. Let x = ) ( - 1 + 2U$) = 0.6944271.... Then the RCF-

expansion of x is x = (0; 1, 2, 3, 1, 2, 44, 2, 1, 3, 2, 1, 1, 10, 1 J. UsingMAPLE we obtained the first 22 partial quotients of the ECF-expansionof x:

Acknowledgements We would like to thank the referee for several helpfulcomments and suggestions.

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Yusuf HARTONO, Cor KRAAIKAMPDelft University of Technology and Thomas Stieltjes Institute for MathematicsITS (CROSS)Mekelweg 42628 CD Delftthe Netherlands

E-mail : y . hartono®its . tudelf t . al , c . okraaikampGits. tudelft onl

Fritz SCHWEIGER

Department of Mathematics, Universitat SalzburgHellbrunnerstr 34A 5020 SalzburgAustriaE-mail : f ritz . s chve iger®mh . sbg . ac . at